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Nonlinear Interaction of Pulses [Elektronische Ressource] / Martina Chirilus-Bruckner. Betreuer: G. Schneider

98 pages
.Nonlinear Interaction of PulsesZur Erlangung des akademischen Grades einesDOKTORS DER NATURWISSENSCHAFTENvon der Fakultät für Mathematik der Universität Karlsruhe (TH) genehmigteDISSERTATIONvonDipl.-Math. techn. Martina Chirilus-Bruckneraus Vatra Dornei, RumänienTag der mündlichen Prüfung: 23. Juli 2009Referent: Prof. Dr. Guido SchneiderKorreferent: Prof. Dr. Michael PlumIntroductionConsider the initial value problem for a nonlinear wave equation, say, the cubic Klein-Gordon equation2 2 3∂ u =∂ u−u+u ,t xwhere x,t,u =u(x,t)∈R, with the initial profile (as depicted in Fig. 0.1) given byu(x,t)| =u (x,t)| ,t=0 pulse t=0∂ u(x,t)| =∂ u (x,t)|t t=0 t pulse t=0with2 ik x+iω t 2 ik x+iω t1 1 2 2u (x,t) =εA (ε(x−c t),ε t)e +εA (ε(x−c t+d),ε t)e +c.c.,pulse 1 1 2 2where c ,k ,ω ,d∈R,0<ε≪ 1, and A (,t) is a localized function for all t∈R.j j j ju(,t)|t=0 c2c1xFigure 0.1: Initial profile given by two localized structures traveling with different velocitiesNow, how does this initial profile evolve in time? How do the localized structures interact? Will acollision destroy them? What if they travel with the same speed? What if we take more than two?Thepresentworkanalyzesallthesenaturalquestionsinthecontextofvariousnonlinearwaveequationswhich arise in applications such as nonlinear fiber optics and photonics, where the localized structuresrepresent light pulses which can be used to transport and process digital data (see Ch. 7).
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Nonlinear Interaction of Pulses
Zur Erlangung des akademischen Grades eines
DOKTORS DER NATURWISSENSCHAFTEN
von der Fakultät für Mathematik der Universität Karlsruhe (TH) genehmigte
DISSERTATION
von
Dipl.-Math. techn. Martina Chirilus-Bruckner
aus Vatra Dornei, Rumänien
Tag der mündlichen Prüfung: 23. Juli 2009
Referent: Prof. Dr. Guido Schneider
Korreferent: Prof. Dr. Michael PlumIntroduction
Consider the initial value problem for a nonlinear wave equation, say, the cubic Klein-Gordon equation
2 2 3∂ u =∂ u−u+u ,t x
where x,t,u =u(x,t)∈R, with the initial profile (as depicted in Fig. 0.1) given by
u(x,t)| =u (x,t)| ,t=0 pulse t=0
∂ u(x,t)| =∂ u (x,t)|t t=0 t pulse t=0
with
2 ik x+iω t 2 ik x+iω t1 1 2 2u (x,t) =εA (ε(x−c t),ε t)e +εA (ε(x−c t+d),ε t)e +c.c.,pulse 1 1 2 2
where c ,k ,ω ,d∈R,0<ε≪ 1, and A (,t) is a localized function for all t∈R.j j j j
u(,t)|t=0 c2
c1
x
Figure 0.1: Initial profile given by two localized structures traveling with different velocities
Now, how does this initial profile evolve in time? How do the localized structures interact? Will a
collision destroy them? What if they travel with the same speed? What if we take more than two?
Thepresentworkanalyzesallthesenaturalquestionsinthecontextofvariousnonlinearwaveequations
which arise in applications such as nonlinear fiber optics and photonics, where the localized structures
represent light pulses which can be used to transport and process digital data (see Ch. 7).
In what follows, we will call these localized structures as “pulses”. They constitute the central theme
of this work.
Brief summary of results
Our main achievement can be stated as follows: By means of perturbation analysis we show that the
two pulses do not interact at all in first order, i.e. they pass through each other undistorted. Hence, atleading order the interaction is elastic, as one would expect from a linear equation. The nonlinearity
shows its influence only at higher orders through interaction effects like a shift of the pulse carrier
wave, a shift of the pulse envelope and a deformation of the pulse shape (see Fig. 2.3- Fig. 2.5).
We give a very precise analytical description of this interaction behavior by means of a reduction of
the original problem to an initial value problem for a
• recursively solvable, decoupled “modulation system” for the envelopes and their higher order
corrections and
• explicit formulas for the interaction effects.
Consequently, we completely separate the description of the internal dynamics of each pulse and the
description ofpure interaction dynamics (well, at least at leading order). At the top of this modulation
system is a Nonlinear Schrödinger equation
2 2∂ A =ν ∂ A +ν |A | A , ν ,ν ∈R,j = 1,2,2 j j,1 1 j j,2 j j j,1 j,1
for each of the two envelopes. The Nonlinear Schrödinger equation has extensively been studied and
is known to exhibit stable, spatially localized solutions — so-called solitons — which can be specified
in terms of elementary functions. This already indicates that the reduced system is much easier to
handle both from an analytical and from a numerical point of view.
Outline of the work
This work is organized in two major parts:
Part I. Concepts of pulse interaction — Separation of internal and interaction dynamics
InChapter 1we give anintroduction to themain analytical toolsthat we makeuseofthroughout
the work. We explain how to derive and rigorously justify the Nonlinear Schrödinger equation
for envelopes of one-pulse solutions for the cubic Klein-Gordon equation. Most of the material
has already been discussed in [KSM92].
Chapter 2 presents a refined perturbation approach that leads to the separation of internal and
interaction dynamics for a multipulse in the setting of a cubic Klein-Gordon equation.
Part II. Pulse interaction in “natural coordinates” and applications
Chapter 3 illustrates a frame work which allows one to transfer the results from the first part of
the work to a more general setting. In fact, this chapter forms the central piece of the present
work.
In the next chapters, i.e. Chapter 4-6, we perform exactly this transfer for the Maxwell-Lorentz
system and a nonlinear wave equation with periodic coefficients. We also revisit the cubic Klein-
Gordon case and reproduce the results from Chapter 2 with the new tools developed in Chapter
3. In this sense, Chapter 4 links Part I and Part II of this work.
Finally, we discuss in Chapter 7 the consequences of the presented results for applications.
Note, that every chapter concludes with a “Chapter Summary” that briefly reviews the main points of
the corresponding chapter.Motivation of the problem
Our research was initiated by the work of Tkeshelashvili et al in [TPB04] where the above problem
was posed for a nonlinear wave equation with periodic coefficients, which is used as a model for light
propagation in nonlinear “photonic crystals” (see Ch. 7).
Photonic crystals consist of a periodic arrangement of materials with different refraction properties.
They are believed to have the ability to support “standing light pulses” which could be used as optical
storage (however, so far, this phenomenon has not been observed experimentally). But how can such
standing pulses be detected? One of the ideas expressed in [TPB04] is that the detection can be real-
ized through a collision with another pulse, which, however, requires a very deep understanding of the
interaction behavior. For this reason [TPB04] contains a formal derivation of explicit formulas for the
leading order interaction effects and a numerical validation of these formulas for special pulse shapes.
Our work is an extension of these results. In particular, we can deal with arbitrary shaped pulses and
derive an explicit formula for an additional interaction effect — the shape deformation.
Another motivation comes from the field of optical communications which is concerned with the trans-
portation of digital data via light pulses (see Ch. 7). To increase the bit rate of the optical fiber that
guides the light pulses, one can simultaneously send pulses on different carrier waves (and, hence, with
different wavelengths), a technique called “Wavelength-Division Multiplexing”. Since optical fibers
exhibit a nonlinear behavior (such as an intensity dependent refraction index), one needs a precise
understanding of the nonlinear interaction both between pulses within one channel and between pulses
from different channels. Such a “Wavelength-Division Multiplexing” scenario can be described by a
multipulse (see Fig. 2.6). In this work we managed to clearly separate the description of the internal
and interaction dynamics associated with such a multipulse and characterize the relation between the
most important parameters that are needed to optimize the multiplexing (see Ch. 2).
Acknowledgements
First and foremost I would like to thank Prof. Dr. Guido Schneider for his extensive advice, support
and encouragement over the last several years and all the fruitful discussions concerning the contents
of my thesis!
I would also like to thank Prof. Michael Plum for agreeing to be a member of my thesis committee
and providing helpful comments and suggestions on my work.
Moreover, I would like to thank my colleague and friend Christopher Chong. I always enjoyed our
discussions not only about mathematics, but also about what it means to be a PhD student!
I thank Prof. Dr. Hannes Uecker for his mathematical advice concerning this thesis.
I am very grateful to Prof. Dr. Kurt Busch and his research group for the stimulating discussions on
the physical problem that initated my research.
Furthermore, I would like to thank Prof. Dr. Willy Dörfler for his constant support.
Finally, I want to thank Simon Friedberger who provided invaluable support during my turbulent time
Aas a PhD student. His assistance with LT X issues is appreciated.E
ThisworkwassupportedbytheGermanResearchFoundation(DFG)throughtheGRK1294: Analysis,
Simulation and Design of Nanotechnological Processes.Contents
I Concepts of pulse interaction — Separation of internal and interaction dy-
namics 11
1 Description of basic tools — One pulse solutions 13
1.1 Formal derivation of modulations equations . . . . . . . . . . . . . . . . . . . . . . . . 14
1.1.1 The ansatz — Approximating pulse solutions . . . . . . . . . . . . . . . . . . . 14
1.1.2 The residual — Measuring the quality of the ansatz . . . . . . . . . . . . . . . 15
1.1.3 Getting the residual small — Derivation of modulation equations . . . . . . . . 15
1.1.4 Canceling higher order harmonics — Algebraic solvability conditions . . . . . . 16
1.1.5 The choice of norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.1.6 Modulations system for one pulse solutions — Internal dynamics . . . . . . . . 17
1.2 Justification of modulation equations for long time scales . . . . . . . . . . . . . . . . . 19
1.3 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Separation of internal and interaction dynamics of multipulse solutions 25
2.1 Interaction of pulses with the same carrier wave - Intrachannel interaction . . . . . . . 25
2.2 Interaction of pulses with different carrier waves - Interchannel interaction . . . . . . . 29
2.2.1 Classical approach - The naive ansatz for a two-pulse . . . . . . . . . . . . . . . 29
2.2.2 Separation of internal and interaction dynamics . . . . . . . . . . . . . . . . . . 32
2.3 Multipulse solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3.1 ε-dependent number of carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
II Pulse interaction in ”natural coordinates” and applications 45
3 Pulse interaction for a system in “natural coordinates” 47
3.1 Transformation to “natural coordinates” . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 Derivation of modulation equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.1 Spectral concentration of pulse solutions . . . . . . . . . . . . . . . . . . . . . . 50
3.2.2 Extended modulation system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.3 Separation of internal and interaction dynamics . . . . . . . . . . . . . . . . . . 59
3.3 Solution strategy for nonlinear wave equations with constant coefficients . . . . . . . . 60
3.4 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 Pulse interaction for a cubic Klein-Gordon equation (revisited) 65
4.1 Transformation to ”natural coordinates” . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1.1 Polar decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.1.2 Diagonalizing transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2 Formal derivation of modulation equations . . . . . . . . . . . . . . . . . . . . . . . . . 674.3 Approximation result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.4 The bridge between x-space and spectral methods. . . . . . . . . . . . . . . . . . . . . 70
4.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5 Pulse interaction for a Maxwell-Lorentz system 71
5.1 Transformation to “natural coordinates” . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.1.1 The system for small k - The problem with zero eigenvalue Jordon blocks . . . 72
5.1.2 The system for|k|→∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.1.3 The diagonalizable system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.1.4 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2 Formal derivation of modulation equations . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3 Approximation result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4 Description of pulse interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6 Pulse interaction for nonlinear wave equations with periodic coefficients 83
6.1 Transformation to “natural coordinates” . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.2 Formal derivation of modulation equations . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2.1 Estimate for the diagonalizing transformation . . . . . . . . . . . . . . . . . . . 87
6.2.2 Getting the residual small in natural coordinates . . . . . . . . . . . . . . . . . 87
6.3 Approximation result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.4 Description of pulse interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7 Applications in nonlinear fiber-optics and photonics 93
7.1 Wavelength-Division multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.2 Detection of standing light pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94List of Figures
0.1 Initial profile given by two localized structures traveling with different velocities . . . . . . . 3
1.1 Apulseconsistingofasinusoidalcarrierwavetravelingwithvelocityc andaspatiallylocalizedph
envelope traveling with velocity c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13gr
2 21.2 Left: The dispersion relation ω =k +1 of the cubic Klein-Gordon equation vs. the scaling in
˜Fourier space of A (dashed); Right: Scaling in x-space of the pulse u . . . . . . . . . . . . . 14
2.1 Initial condition representing intrachannel interaction . . . . . . . . . . . . . . . . . . . . . 26
2.2 Initial condition representing interchannel interaction . . . . . . . . . . . . . . . . . . . . . 29
2.3 Illustration of the carrier shift after collision. Top: The two-pulse after interaction
(solid)vs. thetwopulsetravelingbythemselves(dashed); Bottom: Thecarrierbelonging
to the slower pulse in the presence of the second pulse (solid) and without it (dashed) 33
2.4 Illustration of the envelope shift after collision. Top: The two-pulse after interaction
(solid) vs. the two pulse traveling by themselves (dashed); Bottom: The envelope be-
longing to the slower pulse in the presence of the second pulse (solid) and without it
(dashed) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5 Illustration of the shape deformation after collision. Top: The two-pulse after interac-
tion (solid) vs. the two pulse traveling by themselves (dashed); Bottom: The envelope
belonging to the slower pulse in the presence of the second pulse (solid) and without it
(dashed) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.6 A multipulse consisting of N pulse trains each consisting of M equidistant pulses . . . . . . . 40
3.1 Left panel: Dispersion relation of a cubic Klein-Gordon equation (1.0.1); Middle panel:
Dispersion relation of a Maxwell-Lorentz system (5.0.1)-(5.0.2); Right panel: Dispersion
relation of a cubic Klein-Gordon equation with periodic coefficients (6.0.1) . . . . . . . 47
3.2 Concentration of spectral content for two-pulse solutions . . . . . . . . . . . . . . . . . 49
7.1 Transport of the digital code 10110 by a modulated optical carrier wave through an
optical fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.2 Multiplexing of several carrier waves characterized by their wavelengths (source [KVH]) 94
7.3 “Photonic Crystal” (source [KIT]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Notation
• All constants that appear which are independent of the perturbation parameter ε are denoted
by C.
2• The slow spatial variable is defined by X :=εx and the slow temporal variable by T :=ε t.
• For vectors V the notation [V] means the n-th coordinate.n R
1 −ikx• Fourier transform is defined byF{u}(k) :=uˆ(k) := u(x)e dx.
2π R
s• Sobolev spaces H are equipped with the normZ 1/2
2 s 2kuk s = (1+|k| ) |uˆ(k)| dk .H
R
s• The spaces C are equipped with the normb
sX
j
skuk = k∂ uk 0, kuk 0 = sup|u(x)|.C x C Cb b b x∈R
j=0
• The expressions ∂ f and ∂ f denote the partial derivative of a function f = f(x,t) w.r.t. thex t
first variable x and the second variable t, respectively.
• For functions f =f(x+g(x)) we will have to distinguish between ∂ f and ∂ f.x 1